~ An Advanced Application of Derivatives ~
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Gale the birthday clown is blowing up a fun spherical shaped balloon at a constant rate of
100 cubic inches per minute. How fast is the radius increasing when the sphere contains
50 cubic inches of birthday cake smelling clown breath? How fast is the surface area
increasing at that time?
The Ladder Theorem
When working with real world applications of advanced mathematics it is
obligatory to involve a ladder at some point, preferably one leaning against a
wall.
Gale is trying to escape from the party from an upstairs window, since he just got paid
and doesn’t want to blow up any more balloons.
Unfortunately, the 20 foot extension ladder he leaned against the wall when he arrived is
now slipping down the side of the house. He notices that the bottom is sliding away from
the house at a constant rate of 2 ft/sec.
How quickly is the top of the ladder falling at the exact moment the base of the ladder is
12 feet away from the house? How many more balloons will Gale have to blow up now
that his escape has failed?
~ An Advanced Application of Derivatives ~
Related Rates Problems in 5 Easy Steps
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Having failed to escape the party, Gale is in the backyard blowing up some more balloons.
The kids are a little upset with Gale though because he keeps letting the balloons fly
away. Gale has noticed that the balloons are flying away at a speed of 12 meters per
second. If Gale launches a balloon at a 45 degree angle with the ground, how fast is the
altitude of the balloon changing?
Annoyed with their clown the parents have assigned him the task of filling up the
birthday boy’s new pool. Gale reads on the box that the cylindrical pool has a radius of 5
feet. If the hose is filling the pool at a rate of 28 cubic feet per minute, how fast is the
depth of the pool water increasing?
~ An Advanced Application of Derivatives ~
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A lot of the balloons got caught up in the electrical wires. The dad is forcing Gale to
climb the telephone pole to get them back. Dad is standing 20 feet back from the base of
the pole because he doesn’t want a clown to fall on him, and he notices that the angle of
elevation from him to Gale is changing at a constant rate of 0.46 radians per minute. At
what rate is Gale climbing when the angle of elevation is 0.75 radians?
Gale is climbing so slowing because he is convinced that he is going to be electrocuted
once he gets to the top of the pole. He is thinking about electricity and in particular
resistors…
A resistor is a two-terminal electronic component that produces a voltage across its terminals that is
proportional to the electric current through it in accordance with Ohm's law:
Resistors are elements of electrical networks and electronic circuits and are ubiquitous in most electronic
equipment. Practical resistors can be made of various compounds and films, as well as resistance wire (wire
made of a high-resistivity alloy, such as nickel-chrome).
The primary characteristics of a resistor are the resistance, the tolerance, the maximum working voltage and
the power rating. Other characteristics include temperature coefficient, noise, and inductance.
He is considering two resistors connected in parallel with resistances R1 and R2
measures in ohms (  ). The total resistance, R, is then given by,
1 1
1
 
R R1 R2
Suppose that R1 is increasing at a rate of 0.4  /min and R2 is decreasing at a rate of
0.7  /min. At what rate is R changing when R1  80 and R2  105 ?
Bet you saw this coming a mile away. Gale is now frozen in fear at the top of the pole.
Falling Objects
h(t )  16t 2  Initial Height
The children have a plan. They push the trampoline over so it is directly below Gale and
convince him that it is safe for him to drop down. Actually, they want to get a video of
Gale falling from the electrical pole and getting hit with a pie on his way down to post on
YouTube. The birthday boy sets up with his new video camera 15 feet away and focuses
on Gale who has now reached a height of 25 feet. The kids are ready with the pies and
will hit him when he is 8 feet from the ground. Now, if the birthday boy is able to bring
his camera’s angle of elevation down and stay in focus at a rate of .39 radians / s will he
get the shot of Gale getting hit with the pies?